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To every topologically transitive Cantor dynamical system $(X, \varphi)$ we
associate a group $T(\varphi)$ acting faithfully by homeomorphism on the real
line. It is defined as the group of homeomorphisms of the suspension flow of
$(X, \varphi)$ which preserve every leaf and acts by dyadic piecewise linear
homeomorphisms in the flow direction. We show that if $(X, \varphi)$ is
minimal, the group $T(\varphi)$ is simple, and if $(X, \varphi)$ is a subshift
the group $T(\varphi)$ is finitely generated. The proofs of these two
statements are short and elementary, providing straightforward examples of
finitely generated simple left-orderable groups. We show that if the system
$(X, \varphi)$ is minimal, every action of the group $T(\varphi)$ on the circle
has a fixed point, providing examples of so called "orderable monsters".
We additionally have the following: for every subshift $(X, \varphi)$ the
group $T(\varphi)$ does not have non-trivial subgroups with Kazhdan's property
(T); for every 查看全文>>